Do ideal gases at zero Kelvin have potential energy?
There are many problems :
$1.$ As pointed out by Olin, gas cannot exist as a gas at $0 K$.
$2.$ In ideal gases, interaction between molecules are absent. Hence, there is no potential energy. Remember that Potential energy always has an additive arbitrary constant.
$3.$ As pointed by Wojciech, you would need (to take}energy to cool that down to $0 K$.
I could have just commented about my point but I do not have enough reputation to comment. :P
The real question is "Can any substance at absolute zero have any potential energy?"
There is potential energy in a substance composed of molecules that contain covalent bonds. If the atoms of these molecules can form lower energy covalent bonds with other atoms in a chemical reaction, the difference in bond energies represents chemical potential energy in the original substance that is released in the chemical reaction. The energy released disperses out of the substance and into the surroundings.
It seems that this potential energy should be present in the substance "even if it had no kinetic energy" - which is a definition of a substance at absolute zero.